In this lab you will learn



Special note

In this lab you will be asked to provide screenshots, and you will not need to provide or run R code for any of the questions. Therefore, in this lab you will work on a Word file rather than our usual RMarkdown worksheet. As such, you do not need to load/install any R packages for this lab.

Also, there is a glossary at the end of this lab to help out with terms you may not be familiar with.



Glossary


library(gridExtra) ## for plotting function grid.arrange()
library(deSolve) ## for predator-prey-resource ODE model
library(magrittr) ## for pipe symbol %<>%
library(metR) ## for geom_arrow()
library(OceanView) ## for vector field plots
library(tidyverse)



“Mutualisms are arguably the most important interaction in nature if we consider all their varied forms. Consider the ubiquitous relationship between mycorrhizal fungi and vascular plants, the association of Rhizobium bacteria with leguminous plants, and the diverse community of bacteria in all mammalian guts, including ours. Or plants and their insect pollinators and avian dispersers. Or fish and their cleaning shrimp. Or ants and their homopteran partners. There are many.” Vandermeer & Goldberg 2013

Figure: Caloplaca marina, the Orange Sea Lichen, a crustose lichen that grows on rocks, by Roger Griffith. Lichens are intimate associations between a fungus and photosynthesizing organism. The algae or cyanobacteria live inside the fungus, and share photsynthate with it. Both often reproduce together, as though they were a single organism.

Lichens are fungi that have discovered agriculture

—-Trevor Goward, UBC


You can refer to the table below for the meaning of the symbols appearing in models of facultative and obligate mutualism throughout this lab.

Symbol Meaning
\(N_i\) Population size of species \(i\).
\(\alpha_i\) Interaction coefficient due to species \(i\). For example, \(\alpha_1\) is the impact of species 1 on species 2. Here, if the coefficient is positive, species 1 facilitates species 2 (meaning species 1 increases the growth rate of species 2), whereas if the coefficient is negative the interaction is negative (either predatory/parasitic/pathogenic or competitive).
\(h_i\) Saturation coefficient. It quantifies the saturation of the interaction \(\alpha_i\). For example, if \(h_1 > 0\), then the facilitation (or competitive) impact of species 1 on species 2 will slow down as the abundance of species 1 increases. (That is, if \(h_1>0\) and the population of species 1 is very large, then the impact of each individual of species 1 on the growth rate of species 2 is negligible.)
\(\phi_i\) Threshold abundance for the mutualism to kick in. For example, if \(\phi_1 = 0.2\), then the abundance of species 1 must be at least 0.2 for it to have a net benefit to species 2. If \(N_1 < \phi_1\), then the impact of species 1 on species 2 is actually negative (parasitic or competitive). This can occur if the mutualistic benefit of species 1 to species 2 does not offset the cost to species 2 when species 1 is rare (for example, mutualistic root fungi have a cost, as the plant must provide it with sugars).


Facultative mutualism

Pollination is a classic example of mutualism. The plant provides food to the pollinator, which in turn helps the plant reproduce. In fact, the large diversity of flowering pants compared to non-flowering plants is widely attributed to their relationships with pollinators. When the pollinator-plant relationships is not exclusive, i.e. when the pollinator has other sources of food and the plant has other means of reproducing (e.g. via selfing, vegetative growth, or other pollinators), this is a case of facultative mutualism. In those cases, both participants of the mutualism can sustain a population without the other, but their mutualism contributes to an increase in their respective carrying capacities. For example, a self-pollinating plant may benefit from the higher fitness of the outcrossed seeds enabled by a pollinating bee, which in turn can maintain more hives due to the extra supply of nectar and pollen.

Another famous example of facultative mutualism is clownfish and sea anemones: the fish drive off predators of the anemone, whose venomous tentacles, to which the fish is immune, protect the clownfish from its own predators. Either one can survive without the other, but their association is mutually beneficial.

Figure: Examples of facultative mutualism: Honey bee (Apis mellifera) on sedum flowers (Hylotelephium), by Frank Mikley. Common clownfish (Amphiprion ocellaris) and Ritter’s sea anemone (Heteractis magnifica), by Jan Derk


A simple model of facultative mutualism assumes logistic growth of both mutualists in isolation, and adds a positive interaction term to both species growth rates.

\[ \frac{dN_1}{dt} = r_1N_1\left(1-\frac{N_1+a_2N_2}{K_1}\right) \\ \frac{dN_2}{dt} = r_2N_2\left(1-\frac{N_2+a_1N_1}{K_2}\right) \]

In the equations above, the constant \(a_2\) quantifies the degree to which species 2 increases the growth rate of species 1, and vice-versa for \(a_1\). In the absence of either species, the other one grows to its carrying capacity. For simplicity, let’s set the \(r's\) and \(K's\) to 1 (more generally, we can think of \(N_1\) and \(N_2\) not as head counts but the proportion of each population size relative to its carrying capacity). With these simplifications, the equations above become

\[ \frac{dN_1}{dt} = N_1\left(1-N_1+a_2N_2\right) \\ \frac{dN_2}{dt} = N_2\left(1-N_2+a_1N_1\right) \tag{1} \]

What are the equilibrium abundances of the model above? After some algebra, we arrive at

\[ N_1^* = \frac{1 + a_2}{1 - a_1 a_2} \hspace{2cm} N_2^* = \frac{1 + a_1}{1 - a_1 a_2} \tag{2} \] Since we are assuming \(a_1 > 0\) and \(a_2 > 0\), this means that \(N_1^* > 1\), \(N_2^*>1\). In words, the mutualism allows each species to grow beyond their carrying capacity.

This result is shown in the plot below


In the plot above, rather than having time on the x-axis and abundances on the y-axis, we are plotting the abundance of species 2 against the abundance of species 1. In other words, the axes are \(N_1\) and \(N_2\). A point in the plot then corresponds to a specific value for the population of each species. This type of plot is called a phase plane. We will be using them a lot in this lab, so it is important that you understand what they are showing.

If we start with some initial value of \(N_1\) and \(N_2\), their populations will change over time until they reach dynamical equilibrium. This is illustrated by the two series of black dots, called trajectories, which start at very different initial conditions (one where both \(N_1\) and \(N_2\) are low and another where \(N_1\) is high and \(N_2\) is low) and converge to the same final (equilibrium) state. (So time is shown implicitly on the plot).

Each arrow shows the direction that a point at the tail of the arrow would follow in the next time step. So collectively the arrows give a good visual sense of what would happen to the system over time given any set of initial abundances. Green arrows show regions of the phase plane where both populations are growing or both are declining, and yellow arrows show regions where one species is growing while the other is declining.

The red and blue lines, called isoclines, mark regions of the phase plane where the population of one the species is not immediately changing. \(N_1\) doesn’t change along the red isoclines, and \(N_2\) doesn’t change along the blue isoclines. (Notice how the axes are isoclines, because if either population is zero, it will remain at zero.) Notice how arrows near the red isoclines are mostly vertical (indicating little to no change in species 1), and arrows near blue isoclines are mostly horizontal (little to no change in species 2). Also notice how the trajectory starting in the bottom-right is initially going left as species 1 is decreasing, but then turns right after crossing the red isocline, showing that species 1 starts to increase at that point in time. Notice how the isoclines separate regions of the phase plane where the flow of its respective population changes direction (from growing to declining or vice-versa).

Note: The isoclines are found by setting \(\frac{dN_1}{dt} = 0\) and \(\frac{dN_2}{dt} = 0\) separately, and solving for \(N_2\) in terms of \(N_1\). In the model above they are straight lines, but is not always the case, as we will see below.


Note

The plot above is showing a lot of information, and if you are confused about any part of the explanation above, this is a good time to ask your TA, since a lot of the remainder of this lab relies on being able to interpret phase planes.


Q1. Given the definition of isocline, something special should happen at the intersections of red and blue isoclines. What is special about these points, and what name do we give them? (You may refer to the Glossary at the bottom of this lab write-up for the terminology.)


Q2. How many such intersections can you count on the phase plane above? What is special about only one of these intersections, in terms of the arrows in their vicinity? What do you notice about the abundances of species 1 and 2 at that intersection?


In summary, the phase plane shows basically everything the model can tell us: we know what would ultimately happen to these two species given any set of initial conditions, we know all the possible outcomes, and we know how many of these possible outcomes are stable. In one sentence, the phase plane above shows that the two mutualists will grow to abundances higher than their carrying capacity.

Notice from Equation (2) that the population sizes of both species in the stable equilibrium depend on the parameters \(a_1\) and \(a_2\), which control the intensity of the mutualism. In the following exercises you will determine what happens as we change the values of \(a_1\) and \(a_2\).


Note

Q3 through Q8 refer to this interactive app depicting the phase plane of our model of facultative mutualism. Once you open it, you will notice sliders and boxes on the left side of the screen that let you change the parameter values in the model.


Note

For Q3 and several other questions in this lab, you will be asked to take a screenshot of a plot and insert it as an image file in your worksheet. The easiest way to do this is to work on a Word file rather than an RMarkdown file for this lab.

Alternatively if you would rather work on the usual RMarkdown worksheet, you can insert images by typing ![optional_caption](name_of_the_image_file.file_extension), that is, an exclamation mark followed by square brackets followed by parentheses with the name of the image file, including its extension, inside the parentheses. Save the image file to the same folder where your worksheet is. You can add an optional caption to the image by typing it inside the square brackets.


Q3. Open the app. Notice that by default \(a_1\) and \(a_2\) are set to 0. This means the species do not interact with each other. a) Referring to Equation 1 above, name the type of dynamics that both species undergo in these circumstances. b) Take a screenshot of the phase plane, which you will attach to your answer to this question in your lab. c) Describe how it compares to the phase plane above in terms of the isoclines and the values of \(N_1\) and \(N_2\) at the stable equilibrium.

Note: Don’t worry about the \(h_1\) and \(h_2\) sliders for now. Notice that you can also try out different trajectories by setting different initial values of \(N_1\) and \(N_2\).


Q4. Notice that you can set the \(a_i\) to positive or negative values. a) If both are negative, what type of ecological interaction does that scenario correspond to? b) if one is positive and the other negative, what type of ecological interaction do we have? c) Set both \(a_i\) to negative values and describe the difference from the no-interaction default scenario in terms of the isoclines and the position of the stable equilibrium. Note: Don’t go too negative, or the equilibrium situation may become qualitatively different, which we’ll discuss later. Take a snapshot of your phase plane in (c) and attach it to your worksheet.


It looks like the effect of mutualism is to increase the equilibrium population sizes of the mutualists. But what happens if the mutualism is too strong?


Q5. Slowly increase \(a_2\) and \(a_1\) above zero and describe what happens to the isoclines and the equilibrium point. What happens when \(a_1\) and \(a_2\) are large enough that the product \(a_1 a_2 = 1\)? (You may need to extend the plotting window for \(N_1\) and \(N_2\) by typing in the boxes at the bottom of the slider menu.) What happens once \(a_1 a_2 > 1\)? (In order to see what happens to the trajectories, you may need to change the initial values of \(N_1\) and \(N_2\) by typing in the corresponding boxes)


Q6. Repeat the experiment above, this time for negative \(a_1\) and \(a_2\). What happens when \(a_1\) and \(a_2\) are so negative that \(a_1 a_2 > 1\)? Can the two competing species coexist under these circumstances?


Q5 and Q6 highlight an interesting phenomenon: strong interactions between species tend to destabilize population dynamics (recall Robert May’s results shown in the Module 6 lecture). Q6 also corroborates what we learned in the competition module, namely that when interspecific competition between two species is stronger than intraspecific competition, the species cannot stably coexist.

We learned from Q5 that when mutualism is too strong, the positive effect of a species on another outweighs the species self-regulation represented in the logistic growth part of the model, and thus both populations grow to infinity. Of course in real life that is impossible. And this is in part because the positive effect of a mutualism necessarily saturates as the mutualist population becomes larger and larger. To take the pollination example, once the number of foraging bees vastly exceeds the number of flowers a plant can produce, adding more bees is not going to increase pollination much further.

We can introduce this idea of diminishing returns into our model by replacing the functional response between the mutualists as follows:

\[ \frac{dN_1}{dt} = N_1\left(1-N_1+\frac{a_{2}}{1+h_{2}N_2}N_2\right) \\ \frac{dN_2}{dt} = N_2\left(1-N_2+\frac{a_{1}}{1+h_{1}N_1}N_1\right) \tag{3} \]

The model above is similar to our original model (Equation 1), except that now as \(N_2\) increases, its contribution to the growth of species 1 saturates, and vice-versa for the positive effect of species 1 on species 2. This saturation is encapsulated in the parameters \(h_2\) and \(h_1\), respectively. The higher the value of these parameters, the more quickly the mutualism saturates.

Q7. Going back to our interactive model of facultative mutualism, set \(a_2 = 1.5\) and \(a_1 = 1\). Is there a finite equilibrium? Now set \(h_2 = 2\) and \(h_1 = 1\). What happens now, in terms of the isoclines and the existence of a finite stable equilibrium? Show your plots.


Q8. Repeat the experiment above, this time using large negative values for the interaction coefficients \(a_1\) and \(a_2\) (ie strong interspecific competition). What do you conclude about the effect of saturating interactions on the stability of species dynamics?


Obligate mutualisms

Some mutualisms have become so close as to be indispensable to its participants. Under these so-called obligate mutualisms, neither mutualist can survive without the other. Arguably one such example is gut flora—microorganisms that live in the digestive tracts of animals, including humans. For example, wood-eating termites cannot digest wood without the bacteria housed in their hindgut. In addition to the well-know digestion services the gut microbiome performs, recent research has identified multiple other functions, including inhibiting pathogens, calibrating the immune system, and even modulating the hormonal balance of the host. These functions become apparent in the many maladies afflicting people with an altered gut microbiome. This has led some authors to view the intestinal microbiota as an accessory organ.


Q9. Antibiotics are one of humanity’s chief discoveries, and their deployment marked a quantum leap in healthcare. However, indiscriminate use of antibiotics can lead to serious problems. Based on the discussion above, discuss the potential dangers of misusing antibiotics.


An intriguing example of obligate mutualism is found in chloroplasts and mitochondria, which are organelles in eukaryotic cells with evolutionary origins in free-living prokaryotic organisms. According to the now widely accepted endosymbiotic theory championed by Lynn Margulis, chloroplasts and mitochondria, which are responsible for essential cellular processes such as photosynthesis and respiration in eukaryotic cells, were once free-living cyanobacteria and proteobacteria which at some point were engulfed by eukaryotic cells and started living inside them as either parasites or surviving prey, and eventually tied their reproductive fate to that of their host. The mutualism became so intimate that we no longer distinguish them as separate organisms.


Figure: Chloroplasts vs Mitochondria, by the Amoeba Sisters. Lynn Margulis (1938-2011), early proponent of endosymbiotic theory who prevailed despite intense opposition (her landmark paper was rejected 15 times!). Photo by Jpedreira


Perhaps the most fascinating example of obligate mutualism involves leafcutter ants (subtribe Attina). The ants do not eat the leaves; rather, they feed them to a garden of fungus which is their actual food source. Neither the ants nor the fungus can survive without the other. These gardens are parasitized by a specialized fungal pathogen, and to deal with that, the ants have a secondary symbiosis with a bacterium. The ants house the bacteria in their exoskeleton, and the bacteria produce antifungal chemicals that prevent the growth of the parasitic mold. In sum, through mutualistic relationships, leafcutter ants have discovered not only agriculture, but also pesticides! And it doesn’t stop here: there is even a yeast that parasitizes the ant-bacteria mutualism by growing on the ant cuticle and sapping nutrients from the bacterium!


Q10. What kind of relationship is there between the yeast that disrupts the ant-bacteria mutualism and the parasitic mold that grows on the ants’ fungus garden?


Figure: Left: Leafcutter ants cut and transport plant material to the colony, by Kathy and Sam. Right: Ants tending their garden. The ants’ exoskeleton is covered in Streptomyces bacteria. The bacteria produce antifungal compounds that specifically target a weed fungus which grows in the ants’ fungal gardens. Photo by Cameron Currie


We can model obligate mutualisms by removing the assumption of logistic growth in the absence of the mutualist. Rather, we assume that each population has negative growth in isolation:

\[ \frac{dN_1}{dt} = N_1(\alpha_2 N_2 - N_1) \\ \frac{dN_2}{dt} = N_2(\alpha_1 N_1 - N_2) \tag{4} \] Now, neither species will be able to survive in the absence of the other.



Note

Q11 through Q14 refer to this interactive app depicting the phase plane of our model of obligate mutualism.


Q11. Open the app by clicking here to an interactive implementation of the model above. By default, \(a_1\) and \(a_2\) are set to small values, such that the obligate mutualism is weak. What happens to the populations of both species?


Q12. What happens to the isoclines as you gradually increase \(a_1\) and \(a_2\) (thus increasing the strength of the mutualism)? As you keep increasing them, what eventually happens to the population size of the two species? Is there a finite stable equilibrium at any point of the experiment? Show your plots.


One feature of mutualisms in nature is that there may be a cost to mutualistic adaptations. Classical example include nectaries, which are organs that secrete nectar to attract pollinators, and fleshy fruits, which attract seed dispersers. These structures are costly to produce, and if the mutualist is too rare, they may incur a net loss for the plant.

How does the cost of mutualism affect the dynamics between the mutualist populations? We can try to answer that question with a simple modification to our model:

\[ \frac{dN_1}{dt} = N_1(\alpha_2 (N_2 - \phi_2) - N_1) \\ \frac{dN_2}{dt} = N_2(\alpha_1 (N_1 - \phi_1) - N_2) \tag{5} \] The only difference from Eq 4 is that we introduced threshold abundances \(\phi_1\) and \(\phi_2\). If \(N_2 < \phi_2\), then the mutualism has a negative effect on species 1, reflecting the fact that the cost to species 1 of participating in the mutualism with species 2 outweighs the benefit when species 2 is too rare. Similarly for the interpretation of \(\phi_1\).


Q13. Going back to our interactive model of obligate mutualism, set \(a_2 = 5, \; a_1 = 4\) and verify that the strong mutualism means that both species abundances grow to infinity regardless of the starting point. Now set \(\phi_2 = 0.3, \; \phi_1 = 0.2\), adding a cost to the mutualism for both species. What does the phase plane look like now, in terms of the isoclines and the arrows? Is the intersection point between the blue and red isoclines an equilibrium? Is it stable? Test a few different initial abundances and watch what happens to the species abundances. Show your plots.

Note: We can equally add these cost-to-mutualism effects to our model of facultative mutualism, with similar consequences for the species dynamics.


You may have noticed that our model of obligate mutualism does not seem to allow a stable equilibrium for our mutualists. Rather, their populations either collapse or grow to infinity. How do we reconcile these results with the existence of obligate mutualisms in nature?

The answer is, the same way we did with strong facultative mutualisms. Namely, by acknowledging the fact that mutualisms must saturate as the mutualist abundances grow. For example, adding more and more friendly bacteria to our gut is not likely to bring more and more health benefits indefinitely (in fact, we don’t know that it brings any added benefit at all).

We thus update our model in a similar fashion as we did for facultative mutualisms:

\[ \frac{dN_1}{dt} = N_1\left(\frac{\alpha_{2}}{1+h_{2}N_2}(N_2-\phi_2) - N_1 \right) \\ \frac{dN_2}{dt} = N_2\left(\frac{\alpha_{1}}{1+h_{1}N_1}(N_1-\phi_1) - N_2 \right) \tag{6} \]

Q14. Rerun the obligate mutualism model with the same parameter values as in Q13, but this time set \(h_2 = 1.8\) and \(h_1 = 1.5\), and describe what happens. How many positive equilibria does the model have, and how many are stable? Will every initial condition lead to positive final abundances? Test your answer by trying out a few different initial \(N_1, \; N_2\). Show your plots.



Asymmetric mutualism

Arguably the majority of mutualisms in nature are asymmetric, ie one of the partners gets more than the other out of the association. The mutualism may even be obligate for one partner and facultative for the other.

Consider the mutualism between plants and mycorrhizal fungi, which are fungi that live symbiotically in the roots of the plant. In a typical mycorrhizal association, the plant provides photosynthetic carbon compounds (sugars) to the fungus, which in turn provides the plant with soil nutrients (nitrogen, phosphorus, etc) that its hyphae can absorb much more efficiently than the plant’s roots. The mutualism is asymmetric because the fungus cannot survive without the plant, whereas the plant may be able to acquire soil nutrients without the fungus, albeit inefficiently.

Mychorrizal associations come in two types. Arbuscular mycorrhizal fungi (AMF) penetrate the cortical cells of the roots, to which the plant responds by growing root structures that facilitate the fungal infection. The plant will also respond to signals from the fungus to allocate sugars to its roots. This highly evolved type of mutualism may have been directly responsible for the colonization of land by plants, and today is found in 80% of vascular plant families.

Image: Visualization of a root tuber colonized by an arbuscular mycorrhizal fungus (purple). Notice how the thread-like mycelia (the hyphal mass) vastly increase the nutrient absorption surface of the plant roots. Animation by Scivit


Unlike AMF, ectomycorrhizal fungi (EF) do not penetrate the root cells, and rather grow in the intercellular spaces, forming a hyphal network that ends up covering the surface of the roots. This hyphal sheath helps the plant acquire water and minerals, and can increase plant tolerance to drought and nutrient limitation.

Figure: Ectomycorrhizal mutualisms are found in 2% of plant species, usually woody plants. Left: White spruce, Picea alba, by Allen McGregor. Ectomycorrhizal fungus growing on the roots of white spruce, by Andre Picard


Because the mutualism with the fungus brings a cost to the plant, the net benefit to the plant of its association with the fungus may be contingent on factors such as the type of plant, soil fertility, and degree of root colonization by the fungus. Hoeksema et al (2010) performed a meta-analysis to determine whether the response of plants to having their roots inoculated with mycorrhizal fungi depended on these factors. Results were as follows:

Figure: Parameter estimates (weighted mean ± SE) of plant response to mycorrhizal inoculation for two explanatory variables: plant functional group, and nitrogen fertilization. The number of studies in each group is shown in parentheses. From Hoeksema et al 2010


Q15. Between plants growing in soil that has been treated with fertilizer (a source of nitrogen) and plants growing in untreated soil, which group would you expect to respond more strongly to inoculation? Do the results above line up with those expectations?


Q16. Nitrogen (N)-fixing plants have bacterial symbionts (rhizobia) that provide an independent supply of nitrogen from the mychorrizal fungi. Comparing N-fixing plants to non-N-fixing plants in the chart above, which group would you expect to respond more positively to inoculation with mycorrhizal fungi? Are the results above consistent with your expectation?


Figure: Soybean (Glycinus max), a legume that obtains its nitrogen directly from the atmosphere thanks to its association with rhizobia—nitrogen-fixing bacteria which live in root nodules specially evolved to house them.



Mutualist or parasite?

One interesting possibility that such an asymmetric arrangement may elicit is that while the plant is always a mutualist to the fungus, the fungus may actually become a parasite to the plant if the carbon cost to the plant of subsidizing the fungus outweighs the benefits.

Consider the case of the common milkweed, Asclepias syriaca. This perennial herb is attacked by many herbivorous insects (for example, it is the only food source of the monarch butterfly), and expresses defensive traits that deter insect damage, including a milky latex that oozes out when the plant is chewed (hence the plant’s name) and leaf toxins such as cardenolides. Those defensive substances are primarily carbon-based, but the plant needs other nutrients to synthesize them, mainly nitrogen and phosphorus. The plant’s roots are commonly colonized with mycorrhizal fungi, among them Scuttelospora pellucida and Glomus etunicatum.

Figure: Common milkweed, by Greentec Nursery. Arbuscular mycorrhiza in the roots of common milkweed.


Vannette and Hunter (2011) proposed the resource exchange model of plant defense (REMPD), whereby the costs and benefits of the mutualism with the fungi will affect the plant’s allocation to growth and defense against herbivores. They tested the REMPD via an experiment where milkweed roots were inoculated with different levels of fungal abundance, and observed subsequent plant growth and production of defense traits. The figure below summarizes their findings.

Figure: Expression of Asclepias syriaca defensive traits when grown under experimental manipulation of fungal inoculum density. The left column illustrates responses to colonization by Scutellospora pellucida, while the right column illustrates responses to colonization by Glomus etunicatum inoculum. Solid lines represent the best-fit linear regression model, dashed lines represent the best-fit quadratic regression model, while dotted and dashed lines represent the nonlinear best fit Michaelis–Menten or negative exponential regression model. Trait means ± 1 SE represented are (a, b) plant biomass, (c, d) latex exudation, and (e, f) total foliar cardenolide concentration.


Q17. Focusing on the dashed lines (quadratic regression), how would you describe the response of the plant to increasing root colonization by a) Scutellospora pellucida, b) Glomus etunicatum?


Q18. How would the REMPD explain the change in the relationship between the plant and S. pellucida as root colonization levels increase?


The results above suggest that the relationship between plants and mycorrhizal fungi is more complex than pure mutualism or pure parasitism, and may even be fluid depending on circumstances. What kind of effect can this fluidity have on the population dynamics of these species?

In the spirit of the REMPD, the model below describes a possible association between a plant \(P\) and a fungus \(F\). The impact of the fungus on the plant has a positive component (due to the nutrient subsidies), which saturates as fungal density increases, and a negative component (due to the carbon costs), which saturates as plant numbers increase. The plant can survive without the fungus, but the fungus cannot survive without the plant.

\[ \begin{eqnarray} \frac{dP}{dt} &=& P(1 - P) + \frac{\alpha FP}{1 + hF} - \frac{\beta FP}{1 + gP} \\ \frac{dF}{dt} &=& \frac{\beta PF}{1 + gP} - mF \tag{7} \end{eqnarray} \]


Let’s parse the model above.

  • The equation for the growth rate of the plant, \(dP/dt\), says that if \(F = 0\) (i.e. no fungus present), the second and third terms are zero, the plant grows logistically and equilibrates at a carrying capacity of 1 (notice that if \(F = 0\) and \(P = 1\) then \(dP/dt = 0\)). In other words, the plant does not need the fungus to survive, as it is capable of acquiring soil nutrients (albeit less efficiently) on its own.
  • If the fungus is present, then it will have two effects on the plant, one positive (second term) and one negative (third term).
    • The positive term, quantified by the coefficient \(\alpha\), comes from the provision of soil nutrients by the fungus to the plant. This facilitation saturates as the population of the fungus increases (meaning the plant gets diminishing returns as its roots become colonized by more and more fungus). Parameter \(h\) quantifies this saturation (the higher the \(h\), the faster the facilitation saturates).
    • The negative term, quantified by parameter \(\beta\), comes from the carbon cost to the plant, as the fungus absorbs sugars from the plant’s root.
  • The equation for the growth rate of the fungus, \(dF/dt\), says that without the plant, the fungus has a negative growth rate (notice that if \(P = 0\) the first term becomes \(\frac{0}{1+0}=0\)). The fungus cannot survive without the plant because the plant is its only source of carbon.
  • If the plant is present, then its carbon subsidies to the fungus are quantified by the parameter \(\beta\) (the same parameter that quantifies the cost to the plant, as there is a transference of carbon from the plant to the fungus). Just like the fungus-to-plant subsidy may saturate as the fungus population increases, this plant-to-fungus subsidy may also saturate as the plant population increases. This saturation is quantified by parameter \(g\).


The table below summarizes our symbols and their meanings in this model of asymmetric mutualism:

Symbol Meaning
\(P\) Population of the plant.
\(F\) Population of the root fungus.
\(\alpha\) Benefit to the plant supplied by the fungus.
\(\beta\) Cost to the plant due to the fungus. Doubles down as the benefit to the fungus.
\(h\) Saturation of the fungus-to-plant nutrient subsidy.
\(g\) Saturation of the plant-to-fungus carbon subsidy.


Suppose \(\alpha = 1, \, \beta = 0.75, \, h = 1, \, g = 1, \, m = 0.35\). When \(P = 0.82\), the contribution of the fungus population to the growth of the plant population has the following shape:

Notice the qualitative correspondence between the curve above and the experimental data obtained by Vannette and Hunter for the impact of S. pellucida on milkweed.


Q19. In the curve above, at which level of fungus abundance is the fungus-plant relationship most beneficial for the plant? At which level of fungus abundance does the relationship become parasitic?


What can we say about the stability of such a system where the effect of the fungus on the plant is fluid, and even the nature of the interaction can shift from mutualism to parasitism depending on the fungus abundance?


Note

Q20 through Q24 refer to this interactive app depicting the phase plane of our model of asymmetric mutualism.


Open the app by clicling here. The default values for the mutualism coefficient \(\alpha\) and the parasitism coefficient \(\beta\) is zero. Under these circumstances, as you can see from the phase plane, the plant grows to its carrying capacity of 1, and the fungus goes extinct.


Q20. Add some parasitism by raising \(\beta\) while keeping \(\alpha = 0\). Under these circumstances, the fungus is a pure parasite as it takes carbon from the plant and offers nothing back. What does the isocline of the plant look like? Can the fungus and the plant both sustain positive abundance? Is the equilibrium stable? Show your plot. (Note: Raise the mortality of the fungus so the outcome is more easily visible. You may also adjust the plotting limits using the boxes at the bottom, to zoom in on the equilibrium point.)


Notice how, depending on your set of parameter values for \(m\) and \(\beta\), you see abundance cycles spiraling towards the intersection of the isoclines.


Q21. Where in the course have we seen abundance cycles between two species? Conceptually, how can you connect the result above with those results back then? (think about the roles of the plant and fungus in this scenario)


Now let’s add the mutualism by raising \(\alpha\). To make sure everyone is looking at the same plot, set the parameters to \(m = 0.35, \beta = 1\) and adjust the plotting window for \(N_2\) to 3, then raise \(\alpha\) from 0 to 0.5 in 0.1 steps.) As you do it slowly, you will notice that the effect of the mutualism is to raise the red isocline, which raises the equilibrium towards higher abundance of the fungus. But the blue isocline stays put, meaning that the plant doesn’t get any more abundant, even though it is now assisted by the fungus!


Q22. How would you explain in words this counterintuitive result?


One way to increase the plant equilibrium abundance is to include saturation of the parasitic effect. That is, we can assume that as the plant grows and produces more and more sugars, the fungus becomes satiated, thus allowing the extra carbon to be actually used by the plant to grow.


Q23. Try the idea above by raising the parasitic satiation parameter \(g\) from its default value of 0. Set \(\beta = 1\), \(m = 0.35\), \(\alpha = 0.4\) and window max for N2 = 3, then raise \(g\) slowly from 0 to 1. Describe what happens to the blue isocline and the equilibrium abundances as you gradually increase \(g\).


It may strike you as intuitive that adding mutualism and “predator satiation” (recall Lab 6) leads to an increase in the equilibrium abundance of both species, which is indeed the case. However, something less wholesome is also happening. Set \(\alpha = 0.1, \; \beta = 0.7, \; h = 0.1, \; g = 0.6, \; m = 0.25\), window max for N2 \(= 10\), intial \(N_1 = 0.2\) and initial \(N_2\) = 4, then slowly raise \(\alpha\) from 0.1 to 0.4. You are then gradually increasing the amount of mutualistic benefit of the fungus to the plant.


Q24. What happens to the cycling between plant and fungus populations as you dial up \(\alpha\) above 0.5? (you may need to increase window max for N2 to see it.) How do you think this boom-and-bust behavior may affect the prospects for persistence of these two species?


Notice how when the cycling becomes wider, species spend more time having very low abundance. In a stochastic context, where abundances may fluctuate around their expected values, this would substantially increase the risk of extinction.

Finally, we can see what happens when we also saturate the mutualistic benefit by dialing up \(h\) from its default value of 0. This means that now as the fungus grows, the amount of nutrient the plant absorbs eventually plateaus. As you slowly raise \(h\), you will notice that we recover a finite equilibrium, and as you raise it further you will notice that the cycling becomes tighter and tighter.

In conclusion, positive-positive feedback interactions (mutualisms) destabilize dynamical systems. An unsaturating (and thus unrealistic) mutualism may have no equilibrium, leading to growth to infinity. A strong saturating mutualism may amplify the cycling caused by the parasitism (positive-negative interactions), which in real life could lead to extinction because of stochastic fluctuations.

These results suggest that mutualisms in nature are either weak or quickly saturating. It may be no coincidence that cases of strong, obligate mutualisms in nature have eventually led to either mergers of the organisms involved (eg mitochondria, chloroplasts) or synchronized reproduction where both participants reproduce together as one (eg lichens, fungus gardens).

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